End-Semester Project

Bivariate Analysis on GDP per capita, Sanitation and Life Expectancy across Nations in 2010

Aman Das [BS2206]

Raj Pratap Singh [BS2219]

Shreyansh Mukhopadhyay [BS2147]

Introduction

Overview

This presentation demonstrates the capabilities of Bivariate Analysis on datasets, to infer relationship between various features of Nations.

  • log of GDP per capita: Logarithm (base \(e\)) of Gross Domestic Product (in $) per citizen. Adjusted for Inflation. [lngdp]
  • Sanitation Access %: Percentage of people using at least basic Sanitation facilities, not shared with other households. [snt]
  • Life Expectancy: The average number of years a newly born child would live, provided current mortality patterns hold. [lfx]

Data

Code 
script.dir <- getSrcDirectory(function(x) {x})
setwd(script.dir)

numerise = function(x){
  x[grepl("k$", x)] <- as.numeric(sub("k$", "", x[grepl("k$", x)]))*10^3
  x <- as.numeric(x)
  return(x)
}

d1_raw = read.csv(file.path(".","Data","gdp.csv"), fileEncoding = 'UTF-8-BOM')
d2_raw = read.csv(file.path(".","Data","sanitation.csv"), fileEncoding = 'UTF-8-BOM')
d3_raw = read.csv(file.path(".","Data","life_expectancy.csv"), fileEncoding = 'UTF-8-BOM')

yearname = "X2010"

d1 = d1_raw[!is.na(numerise(d1_raw[, yearname])),][,c("country", yearname)]
colnames(d1)[2] = "lngdp"
d2 = d2_raw[!is.na(numerise(d2_raw[, yearname])),][,c("country", yearname)]
colnames(d2)[2] = "snt"
d3 = d3_raw[!is.na(numerise(d3_raw[, yearname])),][,c("country", yearname)]
colnames(d3)[2] = "lfx"

dtemp = merge(x = d1, y = d2, by = "country")
d = merge(x = dtemp, y = d3, by = "country")

d$lngdp = log(numerise(d$lngdp))

write.csv(d, "./Data/assembled.csv")

kable(head(d, 6L))
country lngdp snt lfx
Afghanistan 6.265301 34.9 60.5
Albania 8.183118 95.2 78.1
Algeria 8.273847 87.0 74.5
Andorra 10.454495 100.0 81.8
Angola 8.291547 41.1 60.2
Antigua and Barbuda 9.546813 86.3 75.9

Univariate Statistics

Measures of Central Tendency

Mean or Arithmetic Mean \(\bar{x}\), Geometric Mean \(\operatorname{GM}(x)\), Harmonic Mean \(\operatorname{HM}(x)\), Median \(\operatorname{median}(x)\) and Mode \(\operatorname{mode}(x)\) are some measures of central tendency in the sample.

\[ \small \begin{aligned} \operatorname{mean}(x)=\bar{x} = \frac{1}{n} \sum _{i=1}^{n}(x_{i}) && \operatorname{GM}(x) = \sqrt[n]{\prod_{i=1}^n} x_i && \operatorname{HM}(x)= n\sum_{i=1}^n x_i^{-1} \end{aligned} \normalsize \] \[ \small \begin{aligned} \operatorname{median}(x)=\begin{cases} x_{(n + 1)/ 2} &: n = 1 \mod{2} \\ \frac{x_{(n/2)} + x_{((n/2)+1)}}{2} &:n = 0 \mod{2} \end{cases} && \operatorname{mode}(x) = x_{(n)} \end{aligned} \normalsize \]

Code 
getmode <- function(v) {
 uniqv <- unique(v)
 freq = max(tabulate(match(v, uniqv)))
 res = uniqv[which.max(tabulate(match(v, uniqv)))]
 if (freq == 1) res = NULL
 return(res)
}

d_central = data.frame(
  row.names = "Variable",
  Variable = c(
    "*ln(GDP)*",
    "*Sanitation*",
    "*Life Exp.*"
  ),
  Mean = c(
    mean(d$lngdp),
    mean(d$snt),
    mean(d$lfx)
  ),
  GM = c(
    geometric.mean(d$lngdp),
    geometric.mean(d$snt),
    geometric.mean(d$lfx)
  ),
  HM = c(
    harmonic.mean(d$lngdp),
    harmonic.mean(d$snt),
    harmonic.mean(d$lfx)
  ),
  Median = c(
    median(d$lngdp),
    median(d$snt),
    median(d$lfx)
  ),
  Mode = c(
    getmode(d$lngdp),
    getmode(d$snt),
    getmode(d$lfx)
  )
)

kable(
  d_central,
  col.names = c(
    "$\\bar{x}$",
    "$\\operatorname{GM}(x)$",
    "$\\operatorname{HM}(x)$",
    "$\\operatorname{median}(x)$",
    "$\\operatorname{mode}(x)$"
  ),
  digits=5
)
\(\bar{x}\) \(\operatorname{GM}(x)\) \(\operatorname{HM}(x)\) \(\operatorname{median}(x)\) \(\operatorname{mode}(x)\)
ln(GDP) 8.54124 8.42229 8.30248 8.48673 9.23014
Sanitation 72.43857 62.58904 47.61862 85.60000 100.00000
Life Exp. 70.54603 69.95538 69.28316 72.40000 73.20000

Measures of Dispersion

Range \(\operatorname{range}(x)\), Semi-Interquartile Range \(\operatorname{SIR}(x)\), Mean Deviation about x’ \(\operatorname{MD}_{(x')}(x)\), Variance \(s_x^2\), Standard Deviation \(s_x\) are some measures of dispersion in the sample.

\[ \small \begin{aligned} \operatorname{Range}(x)=|x_{(n)} - x_{(1)}| && \ Q_1 = \operatorname{median}(x_{(1)}, \ldots ,x_{(\lfloor \frac{n+1}{2} \rfloor)}) && \ Q_3 = \operatorname{median}(x_{(\lfloor \frac{n+2}{2} \rfloor)}, \ldots , x_{(n)}) \end{aligned} \normalsize \] \[ \small \begin{aligned} \operatorname{MD}_{(x')}(x) = \operatorname{mean}(|x_i-x'|) && \operatorname{SIR}(x)=\frac{|Q_1-Q_3|}{2} && s_x = \sqrt{\operatorname{mean}([x_i - \bar{x}]^2)} && s^2_x= (s_x)^2 \end{aligned} \normalsize \]

Code 
getmd = function(x, center = mean(x)){
  md = mean(
    abs(
      x - rep(center, length(x))
      )
    )
  return(md)
}
d_disp = data.frame(
  row.names = "Variable",
  Variable = c(
    "*ln(GDP)*",
    "*Sanitation*",
    "*Life Exp.*"
  ),
  Range = c(
    max(d$lngdp) - min(d$lngdp),
    max(d$snt) - min(d$snt),
    max(d$lfx) - min(d$lfx)
  ),
  SIR = c(
    IQR(d$lngdp)/2,
    IQR(d$snt)/2,
    IQR(d$lfx)/2
  ),
  MD = c(
    getmd(d$lngdp),
    getmd(d$snt),
    getmd(d$lfx)
  ),
  variance = c(
    (sd(d$lngdp))^2,
    (sd(d$snt))^2,
    (sd(d$lfx))^2
  ),
  SD = c(
    sd(d$lngdp),
    sd(d$snt),
    sd(d$lfx)
  )
)

kable(
  d_disp,
  col.names = c(
    "$\\operatorname{Range}(x)$",
    "$\\operatorname{SIR}(x)$",
    "$\\operatorname{MD}_{(\\bar{x})}(x)$",
    "$\\quad \\quad \\quad \\quad s_x^2$",
    "$\\quad \\quad \\quad \\quad s_x$"
  ),
  digits=5
)
\(\operatorname{Range}(x)\) \(\operatorname{SIR}(x)\) \(\operatorname{MD}_{(\bar{x})}(x)\) \(\quad \quad \quad \quad s_x^2\) \(\quad \quad \quad \quad s_x\)
ln(GDP) 6.04435 1.06914 1.17229 2.01791 1.42053
Sanitation 94.03000 24.65000 25.50487 872.29346 29.53461
Life Exp. 50.80000 6.00000 6.98712 75.33494 8.67957

Measures of Shape

Coefficients of Skewness \(g_1\), Kurtosis \(g_2\) describe the symmetry and extremity of tails of the distribution.

\[ \begin{aligned} m_k = \operatorname{mean}([x-\bar{x}]^k) && g_1 = \frac{m_3}{{m_2}^{\frac{3}{2}}} && g_2 = \frac{m_4}{{m_2}^2} \end{aligned} \]

Code 
d_shape = data.frame(
  row.names = "Variable",
  Variable = c(
    "*ln(GDP)*",
    "*Sanitation*",
    "*Life Exp.*"
  ),
  Skewness = c(
    skewness(d$lngdp),
    skewness(d$snt),
    skewness(d$lfx)
  ),
  Kurtosis = c(
    kurtosis(d$lngdp),
    kurtosis(d$snt),
    kurtosis(d$lfx)
  )
)

kable(
  d_shape,
  col.names = c(
    "$g_1$",
    "$g_2$"
  ),
  digits=5
)
\(g_1\) \(g_2\)
ln(GDP) 0.09619 2.14435
Sanitation -0.85989 2.27111
Life Exp. -0.87903 4.02465

Density Plot

Density plots help us in estimating the form, location and spread of the distribution.

Code 
labelfunction = function(val1){
  return(list(c(
    "log of GDP per capita",
    "Sanitation Access %",
    "Life Expectancy"
  )))
}
ggplot(stack(d[2:4]), mapping = aes(x = values))+
geom_density(aes(fill=ind), alpha=0.8)+
labs(
  x=NULL,
  y=NULL
  )+
mytheme+
mycolor+
facet_wrap(~ind, scales="free", labeller = labelfunction, ncol=1)+
  theme(legend.position="none",
        strip.text.x = element_text(size = rel(1.5))
  )

Box Plot

Box plots help us detect potential outliers. They also help us in estimating location and skewness of the distribution.

Code 
labelfunction = function(val1){
  return(list(c(
    "log of GDP per capita",
    "Sanitation Access %",
    "Life Expectancy"
  )))
}
ggplot(stack(d[2:4]), mapping = aes(y = values))+
geom_boxplot(aes(fill=ind, outlier.color = ind), alpha=0.8)+
labs(
  x=NULL,
  y=NULL
  )+
mytheme+
mycolor+
facet_wrap(~ind, scales="free", labeller = labelfunction)+
  theme(axis.text.x=element_blank(),
        legend.position="none",
        strip.text.x = element_text(size = rel(1.5)),
        panel.grid.minor.x = element_blank(),
        panel.grid.major.x = element_blank()
  )

Inferences

Scatter Plot

A Scatter plot is a type of Plot using Cartesian coordinate system to display values for two variables for a set of data. It helps us:

  • take a short glance at effect of two variables.
  • estimate kinds of correlations between variables.
  • infer the direction of correlation (if any).
Code 
sctrplot = function(
    d, x_map, y_map,
    x_lab=waiver(), y_lab=waiver(),
    title=waiver()
){
  plot1 = ggplot(d, mapping = aes(x = x_map, y = y_map))+
    geom_point(
      alpha=0.6
    )+
    mytheme+
    labs(
      x=x_lab,
      y=y_lab,
      title=title
    )
  
  return(plot1)
}

Sanitation vs. GDP per Capita

Life Expectancy vs. GDP per Capita

Life Expectation vs. Sanitation

Inferences

seems like Linear correlation

Bivariate Statistics

Covariance and Correlation Matrices

Covariance \(\operatorname{cov}(x, y)\) is a measure of the joint variability of two random variables \(x\), \(y\).

Correlation is any relationship, causal or spurious, between two random variables \(x\), \(y\). Pearson’s correlation coefficient \(r\) estimates the linear correlation.

\[ \begin{aligned} \operatorname {cov} (x,y)={\frac {\sum _{i=1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{n}} && r_{x,y}= \frac{\operatorname{cov}(x,y)}{s_x s_x} \end{aligned} \]

Code 
cov_mat = cov(d[, 2:4])

kable(cov_mat, digits=5)
lngdp snt lfx
lngdp 2.01791 33.84045 9.52202
snt 33.84045 872.29346 208.54155
lfx 9.52202 208.54155 75.33494

\[A_{i,j} = \operatorname{cov}(x_i, x_j)\]

Code 
cor_mat = cor(d[, 2:4])

kable(cor_mat, digits=5)
lngdp snt lfx
lngdp 1.00000 0.80659 0.77229
snt 0.80659 1.00000 0.81351
lfx 0.77229 0.81351 1.00000

\[A_{i,j} = r_{x_i, x_j}\]

Other Correlation Coefficients

Spearman’s \(r_s\), and Kendall’s \(\tau\) are some other correlation coefficients besides Pearson’s \(r\)

Code 
d_cor = data.frame(
  row.names = "Variable",
  Variable = c(
    "*Sanitation vs. ln(GDP)*",
    "*Life Exp. vs. ln(GDP)*",
    "*Life Exp. vs. Sanitation*"
  ),
  Pearson = c(
    cor(d$snt, d$lngdp, method="pearson"),
    cor(d$lfx, d$lngdp, method="pearson"),
    cor(d$lfx, d$snt, method="pearson")
  ),
  Spearman = c(
    cor(d$snt, d$lngdp, method="spearman"),
    cor(d$lfx, d$lngdp, method="spearman"),
    cor(d$lfx, d$snt, method="spearman")
  ),
  Kendall = c(
    cor(d$snt, d$lngdp, method="kendall"),
    cor(d$lfx, d$lngdp, method="kendall"),
    cor(d$lfx, d$snt, method="kendall")
  )
)

kable(
  d_cor,
  digit = 5,
  col.names = c(
    "*Pearson's* $r$",
    "*Spearman's* $r_s$",
    "*Kendall's* $\\tau$"
  )
)
Pearson’s \(r\) Spearman’s \(r_s\) Kendall’s \(\tau\)
Sanitation vs. ln(GDP) 0.80659 0.85920 0.67458
Life Exp. vs. ln(GDP) 0.77229 0.81639 0.62168
Life Exp. vs. Sanitation 0.81351 0.83513 0.63744

Inferences

Good linear correlation lets try to observe line of best fit.

Linear Regression

Simple Univariate Linear Regression is a method for estimating the relationship \(y_i=f(x_i)\) of a response variable \(y\) with a predictor variable \(x\), as a line that closely fits the \(y\) vs. \(x\) scatter plot.

\[ y_i = \hat{a} + \hat{b} x_i + e_i \]

Where \(\hat{a}\) is the intercept, \(\hat{b}\) is the slope, and \(e_i\) is the ith residual error. We aim to minimize \(e_i\) for better fit.

Ordinary Least Squares

Ordinary Least squares method reduces \(e_i\) by minimizing error sum of squares \(\sum{e_i^2}\).

Code 
olssmry = function(
    d, x_map, y_map,
    x_lab=waiver(), y_lab=waiver(),
    title=waiver()
){
  model = lm(formula=y_map~x_map)
  smry = summary(model, signif.stars=TRUE)
  
  smryvec = c(
    as.numeric(model$coefficients["(Intercept)"]),
    as.numeric(model$coefficients["x_map"]),
    smry$r.squared
  )
  
  return(smryvec)
}

olstab = t(data.frame(
  SvG = olssmry(d, d$lngdp, d$snt),
  LvG = olssmry(d, d$lngdp, d$lfx),
  LvS = olssmry(d, d$snt, d$lfx)
))

colnames(olstab) = c(
  "a", "b",
  "r2"
)

row.names(olstab) = c(
  "*Sanitation vs. ln(GDP)*",
  "*Life Exp. vs. ln(GDP)*",
  "*Life Exp. vs. Sanitation*"
)

olstab$eqn = paste(rep("$y_i$ =", 3), olstab[,"a"], olstab[,"b"])

kable(
  olstab,
  digit = 5
)
x
-70.79844
x
30.24203
x
53.22795
x
16.77006
x
4.71876
x
0.23907
x
0.65059
x
0.59643
x
0.6618
x
\(y_i\) = -70.798436383558 16.7700574017229
\(y_i\) = 30.2420343634012 4.71875501973708
\(y_i\) = 53.227947061323 0.239072697641274

Least Absolute Deviation

Least absolute Deviation method reduces \(e_i\) by minimizing the sum of absolute deviations \(\sum{|e_i|}\).

Code 
ladsmry = function(
    d, x_map, y_map,
    x_lab=waiver(), y_lab=waiver(),
    title=waiver()
){
  model = rq(formula=y_map~x_map)
  smry = summary(model)
  
  smryvec = c(
    as.numeric(model$coefficients[1]),
    as.numeric(model$coefficients[2])
  )
  
  return(smryvec)
}

olstab = t(data.frame(
  SvG = ladsmry(d, d$lngdp, d$snt),
  LvG = ladsmry(d, d$lngdp, d$lfx),
  LvS = ladsmry(d, d$snt, d$lfx)
))

row.names(olstab) = c(
  "*Sanitation vs. ln(GDP)*",
  "*Life Exp. vs. ln(GDP)*",
  "*Life Exp. vs. Sanitation*"
)

kable(
  olstab,
  digit = 5,
  col.names=c(
  "$\\hat{a}$",
  "$\\hat{b}$"
  )
)
\(\hat{a}\) \(\hat{b}\)
Sanitation vs. ln(GDP) -71.23153 16.80472
Life Exp. vs. ln(GDP) 31.99047 4.61340
Life Exp. vs. Sanitation 53.73041 0.23963

Line Fitting

Plotting the estimated Linear Model on the Scatter Plot.

Code 
linearplot = function(
    d, x_map, y_map,
    x_lab=waiver(), y_lab=waiver(),
    title=waiver()
){
  plot1 = ggplot(d, mapping = aes(x = x_map, y = y_map))+
    geom_point(
      alpha=0.6
    )+
    mytheme+
    labs(
      x=x_lab,
      y=y_lab,
      title=title
    )+
    geom_smooth(
      method="lm",
      formula=y~x,
      se=FALSE,
      aes(color = "Ordinary Least Squares")
    )+
    geom_smooth(
      method="rq",
      formula=y~x,
      se=FALSE,
      aes(color = "Least Absolute Deviation")
    )+
    labs(
      color="Linear Model"
    )+
    mycolor
  
  return(plot1)
}

Sanitation vs. GDP per Capita

Life Expectancy vs. GDP per Capita

Life Expectancy vs. Sanitation

Inferences

Partial Correlation

Partial

\[ \begin{aligned} x_i = \hat{a_x} + \hat{b_x} z_i + e_{x,i} && y_i = \hat{a_y} + \hat{b_y} z_i + e_{y,i} \end{aligned} \\ \begin{aligned} r_{x,y;z} = r_{e_{x}, e_{y}} \end{aligned} \]

Partial Correlation
Sanitation vs. ln(GDP) 0.4826925
Life Exp. vs. ln(GDP) 0.3377892
Life Exp. vs. Sanitation 0.5075384

Conclusion